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IIT JAM MA
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Multivariable Calculus
List of top Multivariable Calculus Questions asked in IIT JAM MA
For a > b > 0, consider
D
=
{
(
x
,
y
,
z
)
β
R
3
:
x
2
+
y
2
+
z
2
β€
a
2
and
x
2
+
y
2
β₯
b
2
}
.
D=\left\{(x,y,z) \in \R^3 :x^2+y^2+z^2 \le a^2\ \text{and } x^2+y^2 \ge b^2\right\}.
D
=
{
(
x
,
y
,
z
)
β
R
3
:
x
2
+
y
2
+
z
2
β€
a
2
and
x
2
+
y
2
β₯
b
2
}
.
Then, the surface area of the boundary of the solid D is
IIT JAM MA - 2024
IIT JAM MA
Multivariable Calculus
Functions of Two or Three Real Variables
Define the function
f
:
R
2
β
R
f: \R^2 β \R
f
:
R
2
β
R
by
f(x, y) = 12xy e
-(2x+3y-2)
.
If (a, b) is the point of local maximum of f, then f(a, b) equals
IIT JAM MA - 2024
IIT JAM MA
Multivariable Calculus
Functions of Two or Three Real Variables
Let
π
=
x
{
(
π₯
,
π¦
)
β
R
2
:
π₯
>
0
,
π¦
>
0
}
,
π = x\left\{(π₯, π¦) β β^2 : π₯ > 0, π¦ > 0\right\} ,
S
=
x
{
(
x
,
y
)
β
R
2
:
x
>
0
,
y
>
0
}
,
and f: S β β be given by
f
(
x
,
y
)
=
2
x
2
+
3
y
2
β
log
β‘
x
β
1
6
log
β‘
y
.
f(x,y)=2x^2+3y^2-\log x-\frac{1}{6}\log y.
f
(
x
,
y
)
=
2
x
2
+
3
y
2
β
lo
g
x
β
6
1
β
lo
g
y
.
Then, which of the following statements is/are TRUE ?
IIT JAM MA - 2024
IIT JAM MA
Multivariable Calculus
Functions of Two or Three Real Variables
The area of the region
R
=
{
(
x
,
y
)
β
R
2
:
0
β€
x
β€
1
,
0
β€
y
β€
1
and
1
4
β€
x
y
β€
1
2
}
R=\left\{(x,y) \in \R^2\ : 0\le x \le1,0 \le y \le 1\ \text{and}\ \frac{1}{4} \le xy \le \frac{1}{2} \right\}
R
=
{
(
x
,
y
)
β
R
2
:
0
β€
x
β€
1
,
0
β€
y
β€
1
and
4
1
β
β€
x
y
β€
2
1
β
}
is __________ (rounded off to two decimal places).
IIT JAM MA - 2024
IIT JAM MA
Multivariable Calculus
Integral Calculus
The value of
lim
β‘
t
β
β
(
(
log
β‘
(
t
2
+
1
t
2
)
)
β
1
β«
1
Ο
t
sin
β‘
2
5
x
x
d
x
)
\lim\limits_{tβ\infin}\left((\log(t^2+\frac{1}{t^2}))^{-1} \int\limits_{1}^{\pi t} \frac{\sin^25x}{x}dx\right)
t
β
β
lim
β
(
(
lo
g
(
t
2
+
t
2
1
β
)
)
β
1
1
β«
Ο
t
β
x
s
i
n
2
5
x
β
d
x
)
equals ___________ (rounded off to two decimal places).
IIT JAM MA - 2024
IIT JAM MA
Multivariable Calculus
Integral Calculus
Let T be the planar region enclosed by the square with vertices at the points (0,1), (1,0), (0, β1) and (β1,0). Then, the value of
β¬
T
(
cos
β‘
(
Ο
(
x
β
y
)
)
β
cos
β‘
(
Ο
(
x
+
y
)
)
)
2
d
x
d
y
\iint\limits_T\left(\cos(\pi(x-y))-\cos(\pi(x+y))\right)^2dx\ dy
T
β¬
β
(
cos
(
Ο
(
x
β
y
))
β
cos
(
Ο
(
x
+
y
))
)
2
d
x
d
y
equals ____________ (rounded off to two decimal places).
IIT JAM MA - 2024
IIT JAM MA
Multivariable Calculus
Integral Calculus
Let
S = {(x, y, z) β
R
3
\R^3
R
3
: x
2
+ y
2
+ z
2
< 1}.
Then, the value of
1
Ο
β
s
(
(
x
β
2
y
+
z
)
2
+
(
2
x
β
y
β
z
)
+
(
x
β
y
+
2
z
)
2
)
d
x
d
y
d
z
\frac{1}{\pi}\iiint_s\left((x-2y+z)^2+(2x-y-z)+(x-y+2z)^2\right)dxdydz
Ο
1
β
β
s
β
(
(
x
β
2
y
+
z
)
2
+
(
2
x
β
y
β
z
)
+
(
x
β
y
+
2
z
)
2
)
d
x
d
y
d
z
equals ________ (rounded off to two decimal places).
IIT JAM MA - 2024
IIT JAM MA
Multivariable Calculus
Integral Calculus
For n β
N
\N
N
, if
a
n
=
1
n
3
+
1
+
2
2
n
3
+
2
+
.
.
.
+
n
2
n
3
+
n
a_n=\frac{1}{n^3+1}+\frac{2^2}{n^3+2}+...+\frac{n^2}{n^3+n}
a
n
β
=
n
3
+
1
1
β
+
n
3
+
2
2
2
β
+
...
+
n
3
+
n
n
2
β
then the sequence
{
a
n
}
n
=
1
β
\left\{a_n\right\}_{n=1}^{\infin}
{
a
n
β
}
n
=
1
β
β
converges to ____________ (rounded off to two decimal places)
IIT JAM MA - 2024
IIT JAM MA
Multivariable Calculus
Integral Calculus
Let c > 0 be such that
β«
0
c
e
s
2
d
s
=
3
\int^c_0e^{s^2}ds=3
β«
0
c
β
e
s
2
d
s
=
3
Then, the value of
β«
0
c
(
β«
0
c
e
x
2
+
y
2
d
y
)
d
x
\int\limits_{0}^c\left(\int\limits^c_0 e^{x^2+y^2}dy\right)dx
0
β«
c
β
(
0
β«
c
β
e
x
2
+
y
2
d
y
)
d
x
equals __________(rounded off to one decimal place).
IIT JAM MA - 2024
IIT JAM MA
Multivariable Calculus
Integral Calculus
Suppose f : (β1, 1) β
R
\R
R
is an infinitely differentiable function such that the series
β
j
=
0
β
a
j
x
j
j
!
\sum\limits_{j=0}^{\infin}a_j\frac{x^j}{j^!}
j
=
0
β
β
β
a
j
β
j
!
x
j
β
converges to f(x) for each x β (β1, 1), where,
a
j
=
β«
0
Ο
/
2
ΞΈ
j
cos
β‘
j
(
tan
β‘
ΞΈ
)
d
ΞΈ
+
β«
Ο
/
2
Ο
(
ΞΈ
β
Ο
)
2
cos
β‘
j
(
tan
β‘
ΞΈ
)
d
ΞΈ
a_j=\int\limits_{0}^{\pi/2}\theta^j\cos^j(\tan\theta)d\theta+\int\limits^{\pi}_{\pi/2}(\theta-\pi)^2\cos^j(\tan\theta)d\theta
a
j
β
=
0
β«
Ο
/2
β
ΞΈ
j
cos
j
(
tan
ΞΈ
)
d
ΞΈ
+
Ο
/2
β«
Ο
β
(
ΞΈ
β
Ο
)
2
cos
j
(
tan
ΞΈ
)
d
ΞΈ
for j β₯ 0. Then
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Integral Calculus
Let f(x, y) =
e
x
2
+
y
2
e^{x^2}+y^2
e
x
2
+
y
2
for (x, y) β
R
2
\R^2
R
2
, and a
n
be the determinant of the matrix
(
β
2
f
β
x
2
β
2
f
β
x
β
y
β
2
f
β
y
β
x
β
2
f
β
y
2
)
\begin{pmatrix} \frac{β^2f}{βx^2} & \frac{β^2f}{βxβy} \\ \frac{β^2f}{βyβx} & \frac{β^2f}{βy^2} \end{pmatrix}
(
β
x
2
β
2
f
β
β
y
β
x
β
2
f
β
β
β
x
β
y
β
2
f
β
β
y
2
β
2
f
β
β
)
evaluated at the point (cos(n),sin(n)). Then the limit
lim
β‘
n
β
β
\lim\limits_{n \rightarrow \infin}
n
β
β
lim
β
a
n
is
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Functions of Two or Three Real Variables
The area of the curved surface
S
=
{
(
x
,
y
,
z
)
β
R
3
:
z
2
=
(
x
β
1
)
2
+
(
y
β
2
)
2
}
S = \left\{ (x, y, z) β \R^3 : z^2 = (x β 1)^2 + (y β 2)^2 \right\}
S
=
{
(
x
,
y
,
z
)
β
R
3
:
z
2
=
(
x
β
1
)
2
+
(
y
β
2
)
2
}
lying between the planes z = 2 and z = 3 is
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Integral Calculus
The sum of the series
β
n
=
1
β
2
n
+
1
(
n
2
+
1
)
(
n
2
+
2
n
+
2
)
\sum\limits_{n=1}^{\infin}\frac{2n+1}{(n^2+1)(n^2+2n+2)}
n
=
1
β
β
β
(
n
2
+
1
)
(
n
2
+
2
n
+
2
)
2
n
+
1
β
is equal to _________.(rounded off to two decimal places)
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Integral Calculus
Let f :
R
2
β
R
\R^2 β \R
R
2
β
R
be defined as follows :
f
(
x
,
y
)
=
{
x
4
y
3
x
6
+
y
6
if
(
x
,
y
)
β
(
0
,
0
)
0
if
(
x
,
y
)
=
(
0
,
0
)
f(x,y)=\begin{cases} \frac{x^4y^3}{x^6+y^6} & \text{if }(x,y) \ne (0,0)\\ 0 & \text{if } (x,y)=(0,0) \end{cases}
f
(
x
,
y
)
=
{
x
6
+
y
6
x
4
y
3
β
0
β
if
(
x
,
y
)
ξ
=
(
0
,
0
)
if
(
x
,
y
)
=
(
0
,
0
)
β
Then
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Functions of Two or Three Real Variables
A subset S β
R
2
\R^2
R
2
is said to be bounded if there is an M > 0 such that |x| β€ M and |y| β€ M for all (x, y) β S. Which of the following subsets of
R
2
\R^2
R
2
is/are bounded ?
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Functions of Two or Three Real Variables
Let
f
(
x
,
y
)
=
β¬
(
u
β
x
2
)
+
(
v
β
y
)
2
β€
1
e
β
(
u
β
x
)
2
+
(
v
β
y
)
2
d
u
d
v
.
f(x,y)=\iint\limits_{(u-x^2)+(v-y)^2 \le 1}e^{-\sqrt{(u-x)^2+(v-y)^2}}du\ dv.
f
(
x
,
y
)
=
(
u
β
x
2
)
+
(
v
β
y
)
2
β€
1
β¬
β
e
β
(
u
β
x
)
2
+
(
v
β
y
)
2
β
d
u
d
v
.
Then
lim
β‘
n
β
β
f
(
n
,
n
2
)
\lim\limits_{n \rightarrow \infin}f(n,n^2)
n
β
β
lim
β
f
(
n
,
n
2
)
is
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Integral Calculus
Let f :
R
2
β
R
\R^2 β \R
R
2
β
R
be the function defined as follows :
f
(
x
,
y
)
=
{
(
x
2
β
1
)
2
cos
β‘
2
(
y
2
(
x
2
β
1
)
2
)
if
x
β
Β±
1
0
if
x
=
Β±
1
f(x,y)=\begin{cases} (x^2-1)^2\cos^2(\frac{y^2}{(x^2-1)^2}) & \text{if }x \ne Β±1 \\ 0 & \text{if } x=Β±1\end{cases}
f
(
x
,
y
)
=
{
(
x
2
β
1
)
2
cos
2
(
(
x
2
β
1
)
2
y
2
β
)
0
β
if
x
ξ
=
Β±
1
if
x
=
Β±
1
β
The number of points of discontinuity of f(x, y) is equal to _________.
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Functions of Two or Three Real Variables
For each t β (0, 1), the surface P
t
in
R
3
\R^3
R
3
is defined by
P
t
=
{
(
x
,
y
,
z
)
:
(
x
2
+
y
2
)
z
=
1
,
t
2
β€
x
2
+
y
2
β€
1
}
.
P_t = \left\{(x, y, z) : (x^2 + y^2 )z = 1, t^2 β€ x^2 + y^2 β€ 1\right\}.
P
t
β
=
{
(
x
,
y
,
z
)
:
(
x
2
+
y
2
)
z
=
1
,
t
2
β€
x
2
+
y
2
β€
1
}
.
Let a
t
β R be the surface area of P
t
. Then
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Functions of Two or Three Real Variables
Let S be the triangular region whose vertices are (0, 0),
(
0
,
Ο
2
)
(0,\frac{\pi}{2})
(
0
,
2
Ο
β
)
and
(
Ο
2
,
0
)
(\frac{\pi}{2},0)
(
2
Ο
β
,
0
)
. The value of
β¬
S
sin
β‘
(
x
)
cos
β‘
(
y
)
d
x
d
y
\iint\limits_S\sin(x)\cos(y)dx\ dy
S
β¬
β
sin
(
x
)
cos
(
y
)
d
x
d
y
is equal to ________. (rounded off to two decimal places)
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Integral Calculus
Let f :
R
2
β
R
2
\R^2 β \R^2
R
2
β
R
2
be defined by f(x, y) = (e
x
cos(y), e
x
sin(y)). Then the number of points in
R
2
\R^2
R
2
that do NOT lie in the range of f is
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Functions of Two or Three Real Variables
Which of the following functions is/are Riemann integrable on [0, 1] ?
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Integral Calculus
The value of
lim
β‘
n
β
β
(
n
β«
0
1
x
n
x
+
1
d
x
)
\lim\limits_{n\rightarrow \infin}\left(n\int\limits^1_0\frac{x^n}{x+1}dx\right)
n
β
β
lim
β
(
n
0
β«
1
β
x
+
1
x
n
β
d
x
)
is equal to __________ . (rounded off to two decimal places)
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Integral Calculus
The limit
lim
β‘
a
β
0
(
β«
0
a
sin
β‘
(
x
2
)
d
x
β«
0
a
(
ln
β‘
(
x
+
1
)
)
2
d
x
)
\lim\limits_{a\rightarrow0}\left(\frac{\int\limits_{0}^{a}\sin(x^2)dx}{\int\limits_{0}^{a}(\ln(x+1))^2dx}\right)
a
β
0
lim
β
β
0
β«
a
β
(
l
n
(
x
+
1
)
)
2
d
x
0
β«
a
β
s
i
n
(
x
2
)
d
x
β
β
is
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Integral Calculus
The value of
β«
0
1
β«
0
1
β
x
cos
β‘
(
x
3
+
y
2
)
d
y
d
x
β
β«
0
1
β«
0
1
β
x
cos
β‘
(
x
3
+
y
2
)
d
x
d
y
\int^1_0\int_0^{1-x}\cos(x^3+y^2)dy\ dx-\int^1_0\int_0^{1-x}\cos(x^3+y^2)dx\ dy
β«
0
1
β
β«
0
1
β
x
β
cos
(
x
3
+
y
2
)
d
y
d
x
β
β«
0
1
β
β«
0
1
β
x
β
cos
(
x
3
+
y
2
)
d
x
d
y
is
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Functions of Two or Three Real Variables
Let V be the volume of the region S β
R
3
\R^3
R
3
defined by
S = {(x, y, z) β
R
3
\R^3
R
3
: xy β€ z β€ 4, 0 β€ x
2
+ y
2
β€ 1}.
Then
V
Ο
\frac{V}{Ο}
Ο
V
β
is equal to ________ . (rounded off to two decimal places)
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Integral Calculus
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